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ROBERT FIELD: Well, we're now
well underway into quantum
00:00:27.440 --> 00:00:30.590
mechanics.
00:00:30.590 --> 00:00:38.510
So a lot of the important
stuff goes by very fast.
00:00:38.510 --> 00:00:42.950
So we represent a quantum
mechanical operator
00:00:42.950 --> 00:00:46.070
with a little hat, and
it means do something
00:00:46.070 --> 00:00:48.720
to the thing on its right.
00:00:48.720 --> 00:00:50.520
And it has to be
a linear operator,
00:00:50.520 --> 00:00:53.670
and you want to be sure you
know what a linear operator does
00:00:53.670 --> 00:00:56.370
and what is not a
linear operator.
00:00:56.370 --> 00:00:58.630
This is an eigenvalue equation.
00:00:58.630 --> 00:01:00.780
So we have some
function which, when
00:01:00.780 --> 00:01:03.120
the operator operates
on it, gives back
00:01:03.120 --> 00:01:05.880
a constant times that function.
00:01:05.880 --> 00:01:09.120
The constant is the
eigenvalue, and the function
00:01:09.120 --> 00:01:12.180
is an eigenfunction
of the operator that
00:01:12.180 --> 00:01:16.860
belongs to this eigenvalue,
and all of quantum mechanics
00:01:16.860 --> 00:01:20.520
can be expressed in terms
of eigenvalue equations.
00:01:20.520 --> 00:01:23.670
It's very important, and you
sort of take it for granted.
00:01:23.670 --> 00:01:28.950
Now, one of the important
things about quantum mechanics
00:01:28.950 --> 00:01:33.600
is that we have to find
a linear operator that
00:01:33.600 --> 00:01:37.090
corresponds to the classically
observable quantities.
00:01:37.090 --> 00:01:40.050
And for x the linear
operator is x,
00:01:40.050 --> 00:01:42.210
and for the momentum
the linear operator
00:01:42.210 --> 00:01:45.150
is minus ih bar partial
with respect to x.
00:01:45.150 --> 00:01:47.260
That should bother you two ways.
00:01:47.260 --> 00:01:51.600
One is the i, and the other
is the partial derivative.
00:01:51.600 --> 00:01:55.380
But when you apply this
operator to functions,
00:01:55.380 --> 00:02:02.070
you discover that out pops
something that has the expected
00:02:02.070 --> 00:02:04.920
behavior of
momentum, and so this
00:02:04.920 --> 00:02:07.590
is in fact the
operator that we're
00:02:07.590 --> 00:02:10.710
going to use for momentum.
00:02:10.710 --> 00:02:14.550
And then there is
a commutation rule.
00:02:14.550 --> 00:02:22.140
This commutation rule, xp
minus px, is equal to this.
00:02:22.140 --> 00:02:27.380
This is really the foundation
of quantum mechanics,
00:02:27.380 --> 00:02:31.900
and as I've said before,
many people derive everything
00:02:31.900 --> 00:02:34.580
from a few commutation rules.
00:02:34.580 --> 00:02:38.140
It's really scary,
but you should
00:02:38.140 --> 00:02:41.890
be able to work out
this commutation
00:02:41.890 --> 00:02:47.950
rule by applying xp minus px
to some arbitrary function.
00:02:47.950 --> 00:02:51.160
And going through the symbolics
should take you about 30
00:02:51.160 --> 00:02:53.380
seconds, or maybe it shouldn't.
00:02:53.380 --> 00:02:56.360
Maybe you can go faster.
00:02:56.360 --> 00:02:56.860
OK.
00:03:00.020 --> 00:03:03.060
We have an operator,
and we often
00:03:03.060 --> 00:03:06.660
want to know what
is the expectation
00:03:06.660 --> 00:03:09.840
value of the particular
function, which
00:03:09.840 --> 00:03:13.440
we could symbolize here, but
it's never done that way.
00:03:13.440 --> 00:03:16.500
So we have some
function, and we want
00:03:16.500 --> 00:03:19.500
to calculate its expectation
value of operator A,
00:03:19.500 --> 00:03:22.870
and this is it.
00:03:22.870 --> 00:03:26.860
And so this is a
normalization integral,
00:03:26.860 --> 00:03:31.330
and this normalization integral
is usually taken for granted,
00:03:31.330 --> 00:03:36.520
because we almost always
work with sets of functions
00:03:36.520 --> 00:03:38.620
which are normalized.
00:03:38.620 --> 00:03:41.800
And so if you convince
yourself that it is,
00:03:41.800 --> 00:03:45.710
in fact, they are
normalized, fine,
00:03:45.710 --> 00:03:48.940
and then this is the thing that
you normally would calculate.
00:03:53.840 --> 00:03:56.810
Then, we went to
our first problem
00:03:56.810 --> 00:04:00.230
in quantum mechanics which
is the free particle,
00:04:00.230 --> 00:04:05.480
and the free particle
has some idiosyncrasies.
00:04:05.480 --> 00:04:09.780
The wave function
for the free particle
00:04:09.780 --> 00:04:18.370
has the form e to the ikx
plus e to the minus ikx,
00:04:18.370 --> 00:04:27.270
and the Hamiltonian
is minus h bar
00:04:27.270 --> 00:04:36.900
squared over 2m second partial
with respect to x plus v0.
00:04:36.900 --> 00:04:48.130
So there's no v0 here, and we
have two different exponentials
00:04:48.130 --> 00:04:53.680
and so is this really going
to be an eigenfunction
00:04:53.680 --> 00:04:54.970
of the Hamiltonian?
00:04:54.970 --> 00:05:01.060
This is really p
squared, and so this
00:05:01.060 --> 00:05:03.280
is going to be an
eigenfunction of p squared too.
00:05:08.950 --> 00:05:11.450
All right, so let's show
a little picture here.
00:05:11.450 --> 00:05:19.530
Here is energy, and this
is v0, and let's say
00:05:19.530 --> 00:05:20.850
this is the 0 of energy.
00:05:43.040 --> 00:05:46.900
What are the
eigenvalues of this?
00:05:46.900 --> 00:05:51.290
What does the Hamiltonian
do to this function?
00:05:55.050 --> 00:06:01.310
Well, in order to do that, you
have to calculate something
00:06:01.310 --> 00:06:06.320
like where you have to calculate
the second derivative of each
00:06:06.320 --> 00:06:09.650
of these terms, and the
second derivative of this term
00:06:09.650 --> 00:06:13.805
brings down a minus k squared.
00:06:13.805 --> 00:06:16.240
Now, the second
derivative of this term
00:06:16.240 --> 00:06:22.970
brings down a minus k squared,
and so the energy eigenvalues
00:06:22.970 --> 00:06:32.270
are going to be given by h bar
squared a squared over 2m plus
00:06:32.270 --> 00:06:33.720
v0.
00:06:33.720 --> 00:06:38.430
So these are the
eigenvalues, eigenfunctions,
00:06:38.430 --> 00:06:44.870
the energies of a free particle,
and they're not quantized.
00:06:44.870 --> 00:06:48.780
Now this v0 is something
that will often trip you up,
00:06:48.780 --> 00:06:52.630
because it's hidden here.
00:06:52.630 --> 00:06:53.340
It's not in here.
00:06:55.770 --> 00:06:56.270
OK.
00:07:00.300 --> 00:07:03.030
I'm going to torture
you with something.
00:07:03.030 --> 00:07:10.000
So why are these
two k's the same?
00:07:10.000 --> 00:07:14.990
What would happen if the
k for the positive term
00:07:14.990 --> 00:07:17.080
were different from the
k for the negative term?
00:07:19.720 --> 00:07:22.051
Simple answer.
00:07:22.051 --> 00:07:22.550
Yeah?
00:07:22.550 --> 00:07:24.476
AUDIENCE: There'd
be two eigenvalues?
00:07:24.476 --> 00:07:25.600
ROBERT FIELD: That's right.
00:07:25.600 --> 00:07:29.355
It wouldn't be an eigenvalue.
00:07:29.355 --> 00:07:31.480
It wouldn't be an eigenfunction
of the Hamiltonian.
00:07:31.480 --> 00:07:33.920
It's a mixture of
two eigenvalues,
00:07:33.920 --> 00:07:36.670
and so that's simple.
00:07:36.670 --> 00:07:42.430
But often we might be dealing
with a potential that's
00:07:42.430 --> 00:07:47.440
not simple like this
but has got complexity.
00:07:50.500 --> 00:07:54.605
So suppose we had a
potential that did this.
00:07:59.270 --> 00:08:05.380
The potential is
constant piecewise,
00:08:05.380 --> 00:08:06.270
and so what do we do?
00:08:09.311 --> 00:08:09.810
Yes?
00:08:09.810 --> 00:08:11.610
AUDIENCE: Break down
the function into pieces
00:08:11.610 --> 00:08:12.901
for each in certain boundaries?
00:08:12.901 --> 00:08:16.710
ROBERT FIELD: Yes, and
that's exactly right.
00:08:16.710 --> 00:08:19.700
You do want to
break it up, but one
00:08:19.700 --> 00:08:21.800
of the things I'm
stressing here is
00:08:21.800 --> 00:08:24.110
that you want to be
able to draw cartoons.
00:08:26.800 --> 00:08:32.820
And so we know that if
we choose an energy here,
00:08:32.820 --> 00:08:39.860
there is a certain momentum, or
a certain kinetic energy here,
00:08:39.860 --> 00:08:45.280
and a different kinetic
energy here, and so somehow,
00:08:45.280 --> 00:08:50.680
what you write for the wave
function will reflect that.
00:08:50.680 --> 00:08:53.650
But now qualitatively,
pictorially,
00:08:53.650 --> 00:08:59.830
if we have a wave function
in this region which
00:08:59.830 --> 00:09:03.010
is oscillating like
this, and it'll
00:09:03.010 --> 00:09:06.790
be oscillating at the same
spatial frequency over here,
00:09:06.790 --> 00:09:08.930
well what's going to
be happening here?
00:09:08.930 --> 00:09:10.930
Is it going to be
oscillating faster or slower?
00:09:14.157 --> 00:09:16.605
AUDIENCE: Faster.
00:09:16.605 --> 00:09:18.520
Faster.
00:09:18.520 --> 00:09:21.120
ROBERT FIELD:
Absolutely, and is it
00:09:21.120 --> 00:09:25.325
going to have amplitude
smaller or larger than here?
00:09:29.600 --> 00:09:31.170
You're going to answer, yes?
00:09:33.740 --> 00:09:38.040
Well, let me do a
thought experiment.
00:09:38.040 --> 00:09:41.240
So I'm going to walk from
one side of the blackboard
00:09:41.240 --> 00:09:44.680
to the other, and I'm going to
walk at a constant velocity.
00:09:44.680 --> 00:09:47.705
Then, I'm going to walk
faster and then back
00:09:47.705 --> 00:09:51.310
to this original velocity.
00:09:51.310 --> 00:09:54.040
So what's the
probability of seeing me
00:09:54.040 --> 00:09:58.750
in the middle region
relative to the edge regions?
00:09:58.750 --> 00:10:00.800
Yes?
00:10:00.800 --> 00:10:02.646
AUDIENCE: It's less.
00:10:02.646 --> 00:10:03.770
ROBERT FIELD: That's right.
00:10:03.770 --> 00:10:06.650
The probability,
local probability,
00:10:06.650 --> 00:10:10.330
is proportional to
1 over the velocity,
00:10:10.330 --> 00:10:12.980
and the wave function
is proportional to 1
00:10:12.980 --> 00:10:15.890
over the square root
of the velocity.
00:10:15.890 --> 00:10:17.970
And the velocity is
related to the momentum,
00:10:17.970 --> 00:10:19.680
and so we have everything.
00:10:19.680 --> 00:10:26.070
So we know that the wave here
will be oscillating faster
00:10:26.070 --> 00:10:28.740
and with lower amplitude.
00:10:28.740 --> 00:10:32.220
This is what I want
you to know, and you'll
00:10:32.220 --> 00:10:34.860
be able to use that
cartoon to solve problems.
00:10:37.420 --> 00:10:40.580
If you understand
what's going on here,
00:10:40.580 --> 00:10:43.090
these pictures
will be equivalent
00:10:43.090 --> 00:10:46.720
to global understanding,
and these pictures
00:10:46.720 --> 00:10:57.480
are also part of semi-classical
quantum mechanics.
00:10:57.480 --> 00:11:01.980
I believe you all know classical
mechanics at least a little,
00:11:01.980 --> 00:11:03.960
enough to be useful.
00:11:03.960 --> 00:11:07.220
And what we want to be able to
do in order to draw pictures
00:11:07.220 --> 00:11:11.130
and to understand stuff is
to insert just enough quantum
00:11:11.130 --> 00:11:14.870
mechanics into classical
mechanics so that it's correct.
00:11:17.640 --> 00:11:21.650
Then, all of a sudden, it
starts to make a lot more sense.
00:11:21.650 --> 00:11:22.150
OK.
00:11:36.110 --> 00:11:44.190
So the particle in a box, well,
we have this sort of situation,
00:11:44.190 --> 00:11:49.830
and we have 0 and a.
00:11:49.830 --> 00:11:56.730
So the length of the box is a,
and the bottom of the box v0
00:11:56.730 --> 00:12:00.392
is 0 for this picture.
00:12:00.392 --> 00:12:02.600
Now, one of the things that
I want you to think about
00:12:02.600 --> 00:12:05.100
is, OK, I understand.
00:12:05.100 --> 00:12:06.920
I've solved this problem.
00:12:06.920 --> 00:12:08.870
I know how to
solve this problem.
00:12:08.870 --> 00:12:10.762
I know how to get
the eigenvalues,
00:12:10.762 --> 00:12:12.470
and I know how to get
the eigenfunctions,
00:12:12.470 --> 00:12:14.970
and I know how to
normalize them.
00:12:14.970 --> 00:12:19.550
Well, suppose I move
the box to the side.
00:12:19.550 --> 00:12:25.970
So I move it from
say b to a plus b.
00:12:25.970 --> 00:12:29.406
So it's the same width, but
it's just in a different place.
00:12:29.406 --> 00:12:30.630
Well, did anything change?
00:12:37.830 --> 00:12:39.960
The only thing that changes
is the wave function,
00:12:39.960 --> 00:12:43.690
because you have to
shift the coordinates.
00:12:43.690 --> 00:12:47.570
What happens if I raise
the box or lower the box?
00:12:47.570 --> 00:12:49.861
Will anything change?
00:12:49.861 --> 00:12:51.690
AUDIENCE: [INAUDIBLE]
00:12:51.690 --> 00:12:52.850
ROBERT FIELD: You're hot.
00:12:52.850 --> 00:12:55.010
AUDIENCE: [INAUDIBLE]
00:12:55.256 --> 00:12:56.256
ROBERT FIELD: I'm sorry?
00:12:56.256 --> 00:12:58.050
AUDIENCE: [INAUDIBLE]
00:13:00.119 --> 00:13:00.910
ROBERT FIELD: Yeah.
00:13:00.910 --> 00:13:06.680
If I move the box so that
v0 is not 0, but v0 is 10.
00:13:06.680 --> 00:13:10.490
AUDIENCE: Then, the weight
function will oscillate slower.
00:13:10.490 --> 00:13:12.095
ROBERT FIELD: No.
00:13:12.095 --> 00:13:14.280
AUDIENCE: [INAUDIBLE]
00:13:14.280 --> 00:13:20.600
ROBERT FIELD: So if you
move the box up in energy,
00:13:20.600 --> 00:13:24.321
the wave function is going
to look exactly the same,
00:13:24.321 --> 00:13:26.820
but the energies are going to
be different by the amount you
00:13:26.820 --> 00:13:30.840
move the box up or down, and
this is really important.
00:13:30.840 --> 00:13:35.580
It may seem trivial to some
of you and really obscure
00:13:35.580 --> 00:13:38.670
to others, but you
really want to be
00:13:38.670 --> 00:13:41.370
able to take these things apart.
00:13:41.370 --> 00:13:44.070
Because that will enable
you to understand them
00:13:44.070 --> 00:13:49.400
in a permanent way, and the
cartoons are really important.
00:13:49.400 --> 00:13:53.480
So if you have the solution
to the particle in a box,
00:13:53.480 --> 00:13:56.010
then it doesn't matter
where the box is.
00:13:56.010 --> 00:13:58.515
You know the solution to
any particle in a box.
00:14:04.950 --> 00:14:05.940
OK.
00:14:05.940 --> 00:14:10.560
There is something that I
meant to talk about briefly,
00:14:10.560 --> 00:14:18.350
but when we write
these solutions-- where
00:14:18.350 --> 00:14:22.210
did the other blackboard go?
00:14:22.210 --> 00:14:25.220
All right, well,
I've hidden it--
00:14:25.220 --> 00:14:29.460
so when we have solutions
like e to the ikx
00:14:29.460 --> 00:14:36.920
and e to the minus ikx, so we
have say a here and b here.
00:14:36.920 --> 00:14:39.903
When we go to normalize
a function like this--
00:14:39.903 --> 00:14:42.050
let's put the plus in here--
00:14:42.050 --> 00:14:49.220
then we write psi star psi dx.
00:14:49.220 --> 00:14:52.430
So psi star would
make this go a star
00:14:52.430 --> 00:14:55.160
and this go to e
to the minus ikx,
00:14:55.160 --> 00:15:00.960
and this go to b star
e to the plus ikx.
00:15:00.960 --> 00:15:02.870
So now, we multiply
things together.
00:15:02.870 --> 00:15:06.530
We get an a, a star which
is the square modulus of a,
00:15:06.530 --> 00:15:09.680
and we get e to the ikx
and e to the minus ikx.
00:15:09.680 --> 00:15:11.480
It's 1.
00:15:11.480 --> 00:15:13.780
This is why we use this form.
00:15:13.780 --> 00:15:18.560
The integrals for things
involving e to the ikx
00:15:18.560 --> 00:15:20.850
are either 1 or 0.
00:15:23.530 --> 00:15:28.650
So if you took e to
the ikx, this term,
00:15:28.650 --> 00:15:30.780
and multiplied it by this
term, you'd get an a,
00:15:30.780 --> 00:15:35.330
b star e to the 2 ikx
integrated over a finite region.
00:15:35.330 --> 00:15:35.850
That's 0.
00:15:38.880 --> 00:15:42.060
So we really like this
exponential notation,
00:15:42.060 --> 00:15:45.180
even if you've been brought
up on sines and cosines,
00:15:45.180 --> 00:15:47.880
and you use the sines and
cosines to impose the boundary
00:15:47.880 --> 00:15:48.995
conditions.
00:16:01.890 --> 00:16:06.670
OK, another challenge.
00:16:06.670 --> 00:16:16.230
So this is v0, and the
only problem is this v0--
00:16:16.230 --> 00:16:21.310
well, it looks like this.
00:16:21.310 --> 00:16:25.760
So this is v1.
00:16:25.760 --> 00:16:31.880
OK, so we have now a
particle in this straight.
00:16:31.880 --> 00:16:35.570
It's a hybrid between
the free particle
00:16:35.570 --> 00:16:36.605
and a particle in a box.
00:16:40.290 --> 00:16:42.150
So suppose we're at
an energy like this.
00:16:45.000 --> 00:16:47.160
What's going to happen?
00:16:47.160 --> 00:16:50.610
Well, everything
that's outside--
00:16:50.610 --> 00:16:54.120
everything that's in the
classically-allowed region,
00:16:54.120 --> 00:16:54.990
we understand.
00:16:54.990 --> 00:17:00.860
We know how to deal with it, but
in here, well, that's OK too.
00:17:00.860 --> 00:17:04.819
But inside this
classically-forbidden region,
00:17:04.819 --> 00:17:06.980
the wave function is going
to behave differently.
00:17:06.980 --> 00:17:11.280
Now, I'm going to
assert something.
00:17:11.280 --> 00:17:12.900
It doesn't have nodes.
00:17:12.900 --> 00:17:14.520
It doesn't oscillate.
00:17:14.520 --> 00:17:16.319
It's either
exponentially decreasing
00:17:16.319 --> 00:17:24.750
or exponentially increasing, and
it will never cross 0, never.
00:17:24.750 --> 00:17:25.670
OK.
00:17:25.670 --> 00:17:32.060
So now, if we're solving
a problem involving
00:17:32.060 --> 00:17:42.300
any kind of 1D potential,
number of nodes.
00:17:58.610 --> 00:18:07.190
So for 2D-bound problems, the
number of nodes starts with 0,
00:18:07.190 --> 00:18:10.070
and it corresponds to
the lowest energy state.
00:18:10.070 --> 00:18:15.450
The next state up has 1 node,
and the next state has 2 nodes.
00:18:15.450 --> 00:18:16.940
So by counting the
nodes, you would
00:18:16.940 --> 00:18:21.890
know what the energy order is of
these eigenvalues which is also
00:18:21.890 --> 00:18:24.080
an extremely useful thing.
00:18:24.080 --> 00:18:26.420
If you're thinking about
it or telling your computer
00:18:26.420 --> 00:18:29.750
to find the 33rd
eigenvalue of something,
00:18:29.750 --> 00:18:34.550
because you just run a
calculation that solves
00:18:34.550 --> 00:18:38.090
for an approximate wave
function, and the 33rd,
00:18:38.090 --> 00:18:43.040
it needs 32 nodes.
00:18:43.040 --> 00:18:47.150
And so the computer says,
oh, thank you, master,
00:18:47.150 --> 00:18:49.572
and here is your wave
function, but you
00:18:49.572 --> 00:18:50.780
have to find the right thing.
00:18:50.780 --> 00:18:51.500
OK.
00:18:51.500 --> 00:19:01.040
Now, here is the picture that
you use to remember everything
00:19:01.040 --> 00:19:02.240
about a particle in a box.
00:19:05.610 --> 00:19:08.030
And the wave function
looks like this,
00:19:08.030 --> 00:19:10.565
and the next wave
function looks like that,
00:19:10.565 --> 00:19:16.270
and the next wave
function looks like this.
00:19:16.270 --> 00:19:24.280
And so no nodes, 1
node, 2 nodes, the nodes
00:19:24.280 --> 00:19:29.530
are symmetrically arranged
in the space available.
00:19:29.530 --> 00:19:34.120
And the lobes on one side of
the node and the other side
00:19:34.120 --> 00:19:38.360
have the same amplitude,
different sine,
00:19:38.360 --> 00:19:40.550
and they're all normalized.
00:19:40.550 --> 00:19:43.790
And so the maximum value
for each of these guys
00:19:43.790 --> 00:19:48.920
is 2 over a square root,
where this is 0 to a.
00:19:51.950 --> 00:19:54.570
So that's a fantastic
simplification,
00:19:54.570 --> 00:19:59.060
and it also reminds
you of Mr. DeBroglie.
00:19:59.060 --> 00:20:01.040
He said, you have
to have an integer
00:20:01.040 --> 00:20:03.770
number of half wavelengths--
00:20:03.770 --> 00:20:06.140
well, for the hydrogen--
an integer number
00:20:06.140 --> 00:20:09.150
of wavelengths around a path.
00:20:09.150 --> 00:20:11.220
And for here, you
need an integer number
00:20:11.220 --> 00:20:12.910
of wavelengths for
that round trip
00:20:12.910 --> 00:20:14.760
which is the same
thing or an integer
00:20:14.760 --> 00:20:18.880
number of half wavelengths.
00:20:18.880 --> 00:20:22.850
That's DeBroglie's
idea, and it enables
00:20:22.850 --> 00:20:25.580
you to say, oh well,
let's see if we
00:20:25.580 --> 00:20:28.640
can use this concept
of wavelength
00:20:28.640 --> 00:20:30.530
to approach general problems.
00:20:38.650 --> 00:20:39.160
OK.
00:20:39.160 --> 00:20:45.520
Well, if you do something
to the potential
00:20:45.520 --> 00:20:52.010
by putting a little thing in
it, well, the wave function
00:20:52.010 --> 00:20:55.660
will oscillate more
slowly in that region,
00:20:55.660 --> 00:21:02.960
and that causes it to be at
a higher or lower energy?
00:21:02.960 --> 00:21:05.520
If it's oscillating
more slowly here,
00:21:05.520 --> 00:21:09.740
it has to make it to an integer
number of half wavelengths,
00:21:09.740 --> 00:21:12.170
and so that means
it pushes it up.
00:21:12.170 --> 00:21:16.120
And if you do this,
it'll push it down,
00:21:16.120 --> 00:21:18.490
and you can do terrible things.
00:21:18.490 --> 00:21:21.280
You can put a delta
function there,
00:21:21.280 --> 00:21:24.370
and now you know
everything qualitatively
00:21:24.370 --> 00:21:26.065
that can happen in a 1D box.
00:21:30.661 --> 00:21:31.160
OK.
00:21:31.160 --> 00:21:33.620
One of the things that
bothers people a lot
00:21:33.620 --> 00:21:37.930
is, OK, so we have
some wave function,
00:21:37.930 --> 00:21:42.590
it's got lots of nodes, and the
particle starts out over here.
00:21:42.590 --> 00:21:45.006
How did it get across the node?
00:21:45.006 --> 00:21:46.380
How does it move
across the node?
00:21:48.980 --> 00:21:51.740
Well, the answer
is it's not moving.
00:21:51.740 --> 00:21:52.300
It's here.
00:21:52.300 --> 00:21:52.690
It's here.
00:21:52.690 --> 00:21:53.189
It's here.
00:21:53.189 --> 00:21:56.340
It's everywhere, and this is
just the probability amplitude.
00:21:56.340 --> 00:22:02.200
There is no motion through
a node, no motion at all.
00:22:02.200 --> 00:22:04.000
We are going to do
time-dependent quantum
00:22:04.000 --> 00:22:07.150
mechanics before too long,
and then there will be motion,
00:22:07.150 --> 00:22:11.710
but that motion is encoded
in a different way.
00:22:14.220 --> 00:22:14.720
OK.
00:22:14.720 --> 00:22:21.110
Another thing, suppose you
have a particle in a box,
00:22:21.110 --> 00:22:24.320
and it's in some state, and
I'm going to draw something
00:22:24.320 --> 00:22:26.750
like this again.
00:22:26.750 --> 00:22:29.300
OK, first of all, one,
two, three, four, five,
00:22:29.300 --> 00:22:31.730
which state is that?
00:22:39.010 --> 00:22:43.081
I got-- the hands are right,
six, it's the sixth eigenstate.
00:22:43.081 --> 00:22:43.580
OK.
00:22:43.580 --> 00:22:45.770
Now, suppose--
nothing is moving.
00:22:45.770 --> 00:22:46.610
Right?
00:22:46.610 --> 00:22:49.250
This is a stationary state.
00:22:49.250 --> 00:22:52.940
How would you
experimentally, in principle,
00:22:52.940 --> 00:23:00.710
determine that the particle
is in this n equals 6 state?
00:23:03.840 --> 00:23:07.530
Now, this can be a completely
fanciful experiment, which
00:23:07.530 --> 00:23:10.740
you would never do,
but you could still
00:23:10.740 --> 00:23:14.490
describe how you would do it
and what it would tell you.
00:23:14.490 --> 00:23:17.420
And so, yes.
00:23:17.420 --> 00:23:19.580
AUDIENCE: Try to find
the n equals 6 to n
00:23:19.580 --> 00:23:22.802
equals 7 transition by
irradiating it or something?
00:23:22.802 --> 00:23:23.510
ROBERT FIELD: OK.
00:23:23.510 --> 00:23:27.260
That's the quantum mechanical--
00:23:27.260 --> 00:23:30.320
I agree, spectroscopy
wins always.
00:23:30.320 --> 00:23:37.780
But if you want to observe
the wave function or something
00:23:37.780 --> 00:23:42.460
related to the wave function,
like the number of nodes,
00:23:42.460 --> 00:23:44.270
what would you do?
00:23:44.270 --> 00:23:48.310
And the reason I'm being
very apologetic about this
00:23:48.310 --> 00:23:50.830
is because it's a
crazy idea, But this
00:23:50.830 --> 00:23:52.575
is a one-dimensional system.
00:23:52.575 --> 00:23:53.075
Right?
00:23:53.075 --> 00:23:57.260
It's in the blackboard, and
so you could stand out here
00:23:57.260 --> 00:24:01.580
and shoot particles at it from
the perpendicular direction
00:24:01.580 --> 00:24:04.070
and collect the number
of times you have a hit.
00:24:07.600 --> 00:24:11.020
And so you would
discover that you
00:24:11.020 --> 00:24:16.180
would measure a probability
distribution which
00:24:16.180 --> 00:24:19.465
had the form--
00:24:25.060 --> 00:24:27.720
well, I can't draw
this properly.
00:24:27.720 --> 00:24:31.380
It's going to have one,
two, three, four, five, six,
00:24:31.380 --> 00:24:37.200
six regions separated by a
gap, and what it's measuring
00:24:37.200 --> 00:24:39.500
is psi 6 squared.
00:24:42.020 --> 00:24:47.930
Well, you can't measure, you
cannot observe a wave function,
00:24:47.930 --> 00:24:51.850
but you can observe a
probability distribution wave
00:24:51.850 --> 00:24:52.750
function squared.
00:24:55.520 --> 00:24:57.840
You can also do a
spectroscopic experiment
00:24:57.840 --> 00:25:01.184
and find out what is the
nature of the Hamiltonian.
00:25:01.184 --> 00:25:03.100
And if you know the
nature of the Hamiltonian,
00:25:03.100 --> 00:25:07.039
you can calculate
the wave function,
00:25:07.039 --> 00:25:08.080
but you can't observe it.
00:25:13.830 --> 00:25:14.330
OK.
00:25:14.330 --> 00:25:19.330
Another thing, this harmonic
oscillator-- this particle
00:25:19.330 --> 00:25:22.784
in a box has a minimum
energy which is not
00:25:22.784 --> 00:25:23.825
at the bottom of the box.
00:25:33.530 --> 00:25:36.355
Well, we have something called
the uncertainty principle.
00:25:42.300 --> 00:25:44.980
Now, I'm just pulling
this out of my pocket,
00:25:44.980 --> 00:25:52.450
but I know that x, p is
equal to i h bar not 0,
00:25:52.450 --> 00:25:57.880
and one can derive some
uncertainty principle by doing
00:25:57.880 --> 00:25:59.830
a little bit more mathematics.
00:25:59.830 --> 00:26:02.860
But basically that
uncertainty principle
00:26:02.860 --> 00:26:20.350
is where sigma x is
expectation value
00:26:20.350 --> 00:26:27.270
of x squared minus expectation
value of x squared square root.
00:26:27.270 --> 00:26:29.800
So if we can calculate
this and calculate that,
00:26:29.800 --> 00:26:33.750
we can calculate
the variance in x,
00:26:33.750 --> 00:26:35.600
and you can calculate
the variance in p.
00:26:35.600 --> 00:26:38.680
That's exact, and
that's what you
00:26:38.680 --> 00:26:48.130
can derive from the computation
rule, but for our purposes,
00:26:48.130 --> 00:26:49.510
we can be really crude.
00:26:49.510 --> 00:26:55.400
And so if I'm in this
state, what is delta x?
00:26:59.590 --> 00:27:02.752
What is the range of
possibilities for x?
00:27:02.752 --> 00:27:04.210
AUDIENCE: The box link?
00:27:04.210 --> 00:27:06.700
ROBERT FIELD: Yeah, a.
00:27:06.700 --> 00:27:12.800
OK, and what is
the possibility--
00:27:12.800 --> 00:27:15.640
what is the
uncertainty in p sub x?
00:27:24.000 --> 00:27:29.340
In an eigenstate, we've got
equal amplitudes going this way
00:27:29.340 --> 00:27:30.180
and going that way.
00:27:33.770 --> 00:27:44.510
So we could just say p sub x
positive minus p sub x negative
00:27:44.510 --> 00:27:45.485
which is 2p.
00:27:48.550 --> 00:27:51.490
That's the uncertainty.
00:27:51.490 --> 00:27:55.120
And if we know what
quantum number we're in we
00:27:55.120 --> 00:28:00.670
know what the expectation
value for p of the momentum is,
00:28:00.670 --> 00:28:10.810
and what we derive is that
delta x delta P is equal to hn.
00:28:15.290 --> 00:28:18.490
You can do that, and maybe
I should ask you to really
00:28:18.490 --> 00:28:19.490
be sure you can do that.
00:28:25.250 --> 00:28:29.620
In seconds, because
you really know
00:28:29.620 --> 00:28:35.020
what the possible values of
momentum or momentum squared
00:28:35.020 --> 00:28:36.775
are for a product of a box.
00:28:39.811 --> 00:28:40.310
OK.
00:28:43.190 --> 00:28:47.690
So why is there zero-point
energy, because if you said,
00:28:47.690 --> 00:28:51.140
I had a level at the
bottom of the box,
00:28:51.140 --> 00:28:55.170
we would have the momentum 0.
00:28:55.170 --> 00:28:58.370
The uncertainty and
the momentum is 0,
00:28:58.370 --> 00:29:00.067
and the product
of the uncertainty
00:29:00.067 --> 00:29:02.150
of the moment times the
product of the uncertainty
00:29:02.150 --> 00:29:06.540
in the position has to
be some finite number,
00:29:06.540 --> 00:29:08.580
and you can't do that here.
00:29:08.580 --> 00:29:11.430
And so this is a simple
illustration of the uncertainty
00:29:11.430 --> 00:29:16.820
principle that you have to have
a non-zero zero-point energy.
00:29:16.820 --> 00:29:18.770
That's true for all
one-dimensional problems.
00:29:23.290 --> 00:29:24.096
OK.
00:29:24.096 --> 00:29:25.680
We've got lots of time.
00:29:33.830 --> 00:29:37.620
One of the beautiful things
about quantum mechanics
00:29:37.620 --> 00:29:41.990
is that if you
solved one problem,
00:29:41.990 --> 00:29:44.110
you could solve a whole
bunch of problems,
00:29:44.110 --> 00:29:47.950
and so to illustrate
that, let's consider
00:29:47.950 --> 00:29:50.893
the 3D particle in a box.
00:29:53.660 --> 00:29:58.760
So for the 3D particle
in a box, the Hamiltonian
00:29:58.760 --> 00:30:03.050
can be written as a little
Hamiltonian for the x degree
00:30:03.050 --> 00:30:06.960
of freedom y and z.
00:30:06.960 --> 00:30:08.270
OK.
00:30:08.270 --> 00:30:12.950
So we have three independent
motions of the particle.
00:30:12.950 --> 00:30:15.590
They're not coupled.
00:30:15.590 --> 00:30:18.604
They could be, and we're
interested in letting
00:30:18.604 --> 00:30:19.270
them be coupled.
00:30:19.270 --> 00:30:24.210
But that's where we start
asking questions about reality,
00:30:24.210 --> 00:30:27.120
and that's where we bring
in perturbation theory.
00:30:27.120 --> 00:30:30.270
But for this, oh,
that's fantastic,
00:30:30.270 --> 00:30:33.460
because I know the
eigenvalues of this operator
00:30:33.460 --> 00:30:36.290
and of this operator,
eigenfunctions,
00:30:36.290 --> 00:30:37.800
and of this operator.
00:30:37.800 --> 00:30:39.480
So the problem is
basically solved
00:30:39.480 --> 00:30:42.840
once you solve the 1D box.
00:30:42.840 --> 00:30:49.240
Now, one proviso, what you can
do this separation completely
00:30:49.240 --> 00:30:59.140
formally as long
as hx, hy commute,
00:30:59.140 --> 00:31:02.000
and basically we say the
x, y, and z directions
00:31:02.000 --> 00:31:06.740
don't interact with each other.
00:31:06.740 --> 00:31:08.940
The particle is
free inside the box.
00:31:08.940 --> 00:31:10.520
It's just encountering walls.
00:31:10.520 --> 00:31:16.490
There are no springs or
anything expressing the number
00:31:16.490 --> 00:31:17.960
of degrees of freedom.
00:31:17.960 --> 00:31:19.670
OK.
00:31:19.670 --> 00:31:24.380
So we have now a wave function,
which is a function of x, y,
00:31:24.380 --> 00:31:26.750
and z, but we can
always write it
00:31:26.750 --> 00:31:36.890
as psi x of x, psi
y of y, psi z of z.
00:31:36.890 --> 00:31:41.490
So it's a product of three
wave functions that we know,
00:31:41.490 --> 00:31:47.920
and the energy is
going to be expressed
00:31:47.920 --> 00:32:25.400
as a function of three quantum
numbers, where the box is
00:32:25.400 --> 00:32:26.840
edge lengths a, b, and c.
00:32:30.110 --> 00:32:32.850
You didn't see me
looking at my notes.
00:32:32.850 --> 00:32:38.380
I'm just taking the solution to
1D box, and I'm multiplying it.
00:32:38.380 --> 00:32:43.970
And so now we have the
particle in a 3D box,
00:32:43.970 --> 00:32:49.020
and this is where the ideal
gas law comes from, but not
00:32:49.020 --> 00:32:51.090
in this course.
00:32:51.090 --> 00:32:53.480
So anyway, this
is a simple thing,
00:32:53.480 --> 00:32:55.850
and the wave functions
are simple as well,
00:32:55.850 --> 00:32:59.600
and you can do all
these fantastic things.
00:33:05.740 --> 00:33:11.220
So there are many problems
like a polyatomic molecule.
00:33:11.220 --> 00:33:15.010
In a polyatomic molecule,
if you have n atoms,
00:33:15.010 --> 00:33:20.280
you have 3n minus 6
vibrational modes.
00:33:20.280 --> 00:33:23.250
You might ask, what
is a vibrational mode?
00:33:23.250 --> 00:33:26.640
Well, are they're
independent motions
00:33:26.640 --> 00:33:33.210
of the atoms that satisfy
the harmonic oscillator
00:33:33.210 --> 00:33:37.410
Hamiltonian, which
we'll come to next time.
00:33:37.410 --> 00:33:44.490
And so we have 3n minus
6 exactly solved problems
00:33:44.490 --> 00:33:48.540
all cohabiting in
one Hamiltonian.
00:33:48.540 --> 00:33:51.300
And then we can say, oh yeah,
we got these oscillators,
00:33:51.300 --> 00:33:53.990
and if I stretch
a particular bond,
00:33:53.990 --> 00:33:58.680
it might affect the force
constant for the bending.
00:33:58.680 --> 00:34:03.720
So we can introduce couplings
between the oscillators,
00:34:03.720 --> 00:34:06.320
and in fact, that's what we
do with perturbation theory.
00:34:06.320 --> 00:34:08.030
That's the whole purpose.
00:34:08.030 --> 00:34:12.110
And with that, we can
describe both the spectrum
00:34:12.110 --> 00:34:16.370
and how the spectrum encodes
the couplings between the modes.
00:34:16.370 --> 00:34:18.800
And also, we can describe
what's called intramolecular
00:34:18.800 --> 00:34:21.500
vibrational
redistribution, which
00:34:21.500 --> 00:34:24.210
happens when you have
a very high density
00:34:24.210 --> 00:34:25.760
of vibrational state.
00:34:25.760 --> 00:34:29.210
Energy moves around, because
all the modes are coupled,
00:34:29.210 --> 00:34:31.610
and so even if
you've plucked one,
00:34:31.610 --> 00:34:33.320
the excitation
would go to others.
00:34:33.320 --> 00:34:37.340
And we can understand that
all using the same formalism
00:34:37.340 --> 00:34:39.170
that we're about to develop.
00:34:52.070 --> 00:34:54.670
All right.
00:34:54.670 --> 00:34:56.760
I'm not using my
notes this time,
00:34:56.760 --> 00:34:59.600
because I think there's
just so much insight,
00:34:59.600 --> 00:35:02.105
so I have to keep checking
to see what I've skipped.
00:35:06.881 --> 00:35:07.380
All right.
00:35:07.380 --> 00:35:21.510
So what I've been saying is
whenever the Hamiltonian can
00:35:21.510 --> 00:35:32.500
be expressed as a sum of
individual Hamiltonians,
00:35:32.500 --> 00:35:36.790
whenever we can write
the Hamiltonian this way,
00:35:36.790 --> 00:35:45.790
we can write the wave
function as a product of wave
00:35:45.790 --> 00:35:59.870
functions for
coordinates, xi i1, 2 N.
00:35:59.870 --> 00:36:11.660
And the energies
will be the sum Ein,
00:36:11.660 --> 00:36:16.085
i equals little ei n sub i.
00:36:16.085 --> 00:36:18.740
I equals 1 to N.
00:36:18.740 --> 00:36:19.970
So this is really easy.
00:36:19.970 --> 00:36:26.780
If we have simply a
Hamiltonian, which
00:36:26.780 --> 00:36:30.500
is a sum of individual
particle Hamiltonians,
00:36:30.500 --> 00:36:32.330
we don't even have
to stop to think.
00:36:32.330 --> 00:36:34.663
We know the wave functions
and the eigenvalues.
00:36:38.310 --> 00:36:38.810
OK.
00:36:44.370 --> 00:36:56.140
Now, suppose the Hamiltonian
is this plus that.
00:36:56.140 --> 00:37:00.370
So here, we have a Hamiltonian,
and this is this simply the
00:37:00.370 --> 00:37:01.750
uncoupled Hamiltonian.
00:37:01.750 --> 00:37:06.340
This is what we'd like nature
to be, but nature isn't so kind,
00:37:06.340 --> 00:37:08.770
and there are some
coupling terms.
00:37:08.770 --> 00:37:13.960
And so we know the
eigenfunctions and eigenvalues
00:37:13.960 --> 00:37:15.100
for this Hamiltonian.
00:37:15.100 --> 00:37:20.830
We call them the basis functions
and the zero-order energies,
00:37:20.830 --> 00:37:22.390
and then there is
this thing that
00:37:22.390 --> 00:37:24.909
couples them and leads
to complications,
00:37:24.909 --> 00:37:26.200
and that's perturbation theory.
00:37:26.200 --> 00:37:27.730
We're going to do that.
00:37:27.730 --> 00:37:28.830
OK.
00:37:28.830 --> 00:37:40.850
So now, let me just say,
on page nine of your notes,
00:37:40.850 --> 00:37:48.420
there's the words next
time, and those are going
00:37:48.420 --> 00:37:50.495
to be replaced by you should.
00:37:54.930 --> 00:37:57.790
There's a whole bunch of things
that I want you to consider,
00:37:57.790 --> 00:38:02.130
and I was planning on
talking about them,
00:38:02.130 --> 00:38:03.434
but they're all pretty trivial.
00:38:03.434 --> 00:38:05.100
And so there are a
whole bunch of things
00:38:05.100 --> 00:38:10.697
you should study, because I will
ask you questions about them.
00:38:13.620 --> 00:38:18.000
And of greatest
importance is the ability
00:38:18.000 --> 00:38:21.351
to calculate things like that.
00:38:25.050 --> 00:38:25.550
OK.
00:38:25.550 --> 00:38:29.810
Now, I'm going to give
you a whole bunch of facts
00:38:29.810 --> 00:38:32.630
which I may not have derived.
00:38:32.630 --> 00:38:37.620
But you're going
to live with them,
00:38:37.620 --> 00:38:39.870
and you can ask me questions.
00:38:39.870 --> 00:38:42.300
Some of these things are
theorems that we can prove,
00:38:42.300 --> 00:38:45.930
but the proof of the
theorem is really boring.
00:38:45.930 --> 00:38:49.330
Understanding what it
is is really wonderful.
00:38:49.330 --> 00:39:01.070
So all eigenfunctions
that belong
00:39:01.070 --> 00:39:05.090
to different eigenvalues--
00:39:09.440 --> 00:39:13.930
of whatever operator we
want, the Hamiltonian,
00:39:13.930 --> 00:39:17.620
some other operator--
00:39:17.620 --> 00:39:20.155
are orthogonal.
00:39:23.340 --> 00:39:27.060
That's a fantastic
simplification.
00:39:27.060 --> 00:39:29.310
So if you have
two eigenfunctions
00:39:29.310 --> 00:39:36.390
of the Hamiltonian, of the
position operator, anything,
00:39:36.390 --> 00:39:39.730
those eigenfunctions
are orthogonal.
00:39:39.730 --> 00:39:46.450
Their integral is 0,
very, very useful.
00:39:46.450 --> 00:39:53.410
Then, one of the
initial postulates
00:39:53.410 --> 00:39:57.670
about quantum
mechanics is this idea
00:39:57.670 --> 00:40:00.810
that the wave functions
are well-behaved.
00:40:04.310 --> 00:40:08.870
Well, if I were to state
it at the beginning,
00:40:08.870 --> 00:40:11.980
you wouldn't know what's
well-behaved and ill-behaved,
00:40:11.980 --> 00:40:12.950
but now I can tell you.
00:40:20.240 --> 00:40:23.360
One of the things is that the
wave function is continuous,
00:40:23.360 --> 00:40:25.000
no matter what the
potential does.
00:40:27.680 --> 00:40:39.640
The derivative is continuous,
except at an infinite barrier.
00:40:39.640 --> 00:40:42.070
So you come along, and you
hit an infinite barrier,
00:40:42.070 --> 00:40:49.400
and you've already seen that
with the particle in a box.
00:40:49.400 --> 00:40:52.330
The wave function is continuous
at the edge of the box,
00:40:52.330 --> 00:40:54.420
but the derivative
is discontinuous,
00:40:54.420 --> 00:40:58.830
and it's because it's
an infinite wall.
00:40:58.830 --> 00:41:01.950
That's a pretty violent thing
to make the first derivative be
00:41:01.950 --> 00:41:03.360
discontinuous.
00:41:03.360 --> 00:41:09.080
The secondary derivative
is continuous,
00:41:09.080 --> 00:41:16.424
except at any sudden
change in the potential.
00:41:20.140 --> 00:41:24.480
So when you're
solving 1D problems,
00:41:24.480 --> 00:41:28.830
and you've got a solution that
works in the various regions,
00:41:28.830 --> 00:41:31.440
and you want to
connect them together,
00:41:31.440 --> 00:41:35.790
these provide some rules
about the boundary conditions.
00:41:40.150 --> 00:41:46.780
So now, most real systems
don't have infinite walls
00:41:46.780 --> 00:41:52.020
or infinitely sharp steps.
00:41:52.020 --> 00:41:56.164
So for calculation of
physically reasonable things,
00:41:56.164 --> 00:41:58.580
wave function for the first
derivative, second derivative,
00:41:58.580 --> 00:42:00.080
they are continuous.
00:42:00.080 --> 00:42:05.340
But for solving a problem,
we like these steps,
00:42:05.340 --> 00:42:09.210
because then we know how to
impose boundary conditions,
00:42:09.210 --> 00:42:14.420
and that gets us a much
easier thing to calculate.
00:42:14.420 --> 00:42:15.020
OK.
00:42:15.020 --> 00:42:20.070
Now-- oh, that's
where it went, OK--
00:42:26.680 --> 00:42:33.200
semi classical
quantum mechanics.
00:42:39.360 --> 00:42:42.420
We know that the
energy classically
00:42:42.420 --> 00:42:44.646
is p squared over 2m.
00:42:44.646 --> 00:42:47.030
Right?
00:42:47.030 --> 00:42:48.980
It's 1/2 mv squared, but
that's the same thing
00:42:48.980 --> 00:42:50.450
as p squared over 2m.
00:42:50.450 --> 00:42:53.340
In quantum mechanics, when
we talk about Hamiltonians,
00:42:53.340 --> 00:42:58.180
the variables are x
and p not x and v.
00:42:58.180 --> 00:43:01.820
So that seems like
a picky thing,
00:43:01.820 --> 00:43:04.280
but it turns out to
be very important.
00:43:04.280 --> 00:43:13.680
And so we can say,
well, the momentum
00:43:13.680 --> 00:43:17.344
can be a function
of x, classically.
00:43:24.610 --> 00:43:32.950
So I just solved this problem,
and if the potential is not
00:43:32.950 --> 00:43:37.480
constant, then the momentum,
classical momentum,
00:43:37.480 --> 00:43:39.000
is not constant.
00:43:39.000 --> 00:43:43.420
But we know what
it is everywhere,
00:43:43.420 --> 00:43:50.880
and we also know that
the wavelength is
00:43:50.880 --> 00:43:52.290
equal to h over p.
00:43:54.970 --> 00:43:59.680
So we could make a step
into the unknown saying,
00:43:59.680 --> 00:44:03.280
well, the wavelength for
a non-constant potential
00:44:03.280 --> 00:44:05.860
is a function of
coordinate, and it's
00:44:05.860 --> 00:44:10.370
going to be equal
to h over p of x.
00:44:10.370 --> 00:44:14.180
That's semi-classical
quantum mechanics, everything
00:44:14.180 --> 00:44:16.230
you would possibly want.
00:44:16.230 --> 00:44:18.110
Now, for one-dimensional
problems,
00:44:18.110 --> 00:44:25.960
you can solve in terms of this
coordinate dependent wavelength
00:44:25.960 --> 00:44:29.900
which is related to
the local momentum.
00:44:29.900 --> 00:44:34.340
And so it doesn't matter how
complicated the problem is,
00:44:34.340 --> 00:44:37.610
you know that you can calculate
the spatial modulation
00:44:37.610 --> 00:44:40.790
frequency, you could
calculate the amplitude,
00:44:40.790 --> 00:44:45.860
is it big or small,
based on these ideas
00:44:45.860 --> 00:44:50.090
of the classical
momentum function.
00:44:50.090 --> 00:44:52.670
So this demonstration
of my walking
00:44:52.670 --> 00:44:59.060
across the room slow, fast, slow
tells you about probability.
00:44:59.060 --> 00:45:05.390
So if you use this formula,
you know the node spacings,
00:45:05.390 --> 00:45:09.340
and you know the amplitudes.
00:45:09.340 --> 00:45:13.770
Now, what you don't know
is where are the nodes?
00:45:13.770 --> 00:45:17.520
You know how far
they are apart, but I
00:45:17.520 --> 00:45:19.230
have to be humble about this.
00:45:19.230 --> 00:45:21.030
In order to calculate
where the nodes are,
00:45:21.030 --> 00:45:25.980
I have to do a little bit more
in order to pin them down.
00:45:25.980 --> 00:45:28.800
But mostly, when you're trying
to understand how something
00:45:28.800 --> 00:45:33.350
works, you want to know the
amplitude of the envelope,
00:45:33.350 --> 00:45:36.180
and that's a
probability, and so it's
00:45:36.180 --> 00:45:40.320
related to 1 over the
square root of the momentum.
00:45:40.320 --> 00:45:42.540
I'm sorry, the amplitude
is 1 over the square root
00:45:42.540 --> 00:45:44.371
of the momentum, and
the nodes spacings,
00:45:44.371 --> 00:45:45.870
those are the things
you want to do,
00:45:45.870 --> 00:45:47.775
and you want to know
them immediately.
00:45:53.230 --> 00:45:56.460
And a couple other facts that I
told you earlier, but I think I
00:45:56.460 --> 00:45:58.360
want to emphasize them--
00:45:58.360 --> 00:46:08.920
the energy order,
number of nodes,
00:46:08.920 --> 00:46:10.090
number of internal nodes.
00:46:14.200 --> 00:46:17.410
For 1D problems, you
never skip a number of--
00:46:17.410 --> 00:46:22.626
so you can't say, there is no
wave function with 13 nodes,
00:46:22.626 --> 00:46:26.310
even if you don't
like being unlucky.
00:46:26.310 --> 00:46:33.470
And it's there, and so if you
want the 13th energy level,
00:46:33.470 --> 00:46:38.170
you want something
with 12 nodes,
00:46:38.170 --> 00:46:40.160
and that also focuses things.
00:46:40.160 --> 00:46:42.800
So these are amazingly
wonderful things,
00:46:42.800 --> 00:46:44.930
because you can
get them from what
00:46:44.930 --> 00:46:48.280
you know about
classical mechanics,
00:46:48.280 --> 00:46:53.750
and it's easy to embed them
into a kind of half quantum
00:46:53.750 --> 00:46:56.080
mechanics.
00:46:56.080 --> 00:46:58.360
And since, I told
you, this course
00:46:58.360 --> 00:47:03.340
is for use and insight, not
admiration of philosophy
00:47:03.340 --> 00:47:07.160
or historical development, and
this is what you want to do.
00:47:07.160 --> 00:47:09.580
You look at the
problem and sketch
00:47:09.580 --> 00:47:12.460
how is the wave
function going to behave
00:47:12.460 --> 00:47:15.010
and perhaps how a
particular thing
00:47:15.010 --> 00:47:16.630
at some place in
the potential is
00:47:16.630 --> 00:47:19.840
going to affect the energy
levels or any other observable
00:47:19.840 --> 00:47:21.080
property.
00:47:21.080 --> 00:47:25.660
And so the cartoons are really
your guide to getting things
00:47:25.660 --> 00:47:28.770
right, but you
really have to invest
00:47:28.770 --> 00:47:35.491
in developing the sense of
how to build these cartoons.
00:47:35.491 --> 00:47:35.990
OK.
00:47:35.990 --> 00:47:37.830
I'm finished early again.
00:47:37.830 --> 00:47:39.260
Does anybody have any questions?
00:47:42.020 --> 00:47:42.520
OK.
00:47:42.520 --> 00:47:46.190
We start the harmonic
oscillator next time.
00:47:46.190 --> 00:47:48.870
OK, so I can say a
couple of things.
00:47:48.870 --> 00:47:52.320
The wave functions,
the solution to the 1D
00:47:52.320 --> 00:47:56.570
box and the free particle,
they're really simple.
00:47:56.570 --> 00:47:58.280
The solution to the
harmonic oscillator
00:47:58.280 --> 00:48:01.740
involves a complicated
differential equation
00:48:01.740 --> 00:48:04.385
which the mathematicians
have solved and worked
00:48:04.385 --> 00:48:07.210
out all the properties.
00:48:07.210 --> 00:48:12.610
But there is a really
important simplification
00:48:12.610 --> 00:48:17.590
that enables us to proceed
with even greater velocity
00:48:17.590 --> 00:48:19.120
in the harmonic
oscillator than we
00:48:19.120 --> 00:48:20.920
would in a particle in a box.
00:48:20.920 --> 00:48:26.320
And they are these
things called a and a
00:48:26.320 --> 00:48:30.370
dagger, creation and
annihilation operators, where
00:48:30.370 --> 00:48:37.450
when we operate on psi with
this creation operator,
00:48:37.450 --> 00:48:48.460
it converts it to square root of
v plus 1, psi v plus 1 further.
00:48:48.460 --> 00:48:51.520
These things, we don't ever
have to do an integral.
00:48:51.520 --> 00:48:53.530
Once you're in harmonic
oscillator land,
00:48:53.530 --> 00:49:00.490
everything you need comes from
these wonderful operators.
00:49:00.490 --> 00:49:02.650
And so even though the
differential equation
00:49:02.650 --> 00:49:07.070
is a little bit
scary for chemists,
00:49:07.070 --> 00:49:10.200
these things make
everything trivial.
00:49:10.200 --> 00:49:14.460
And so we use the
harmonic oscillator,
00:49:14.460 --> 00:49:16.910
and the particle in
a box to illustrate
00:49:16.910 --> 00:49:19.440
time-dependent
quantum mechanics.
00:49:19.440 --> 00:49:24.840
They each have their own special
advantages for simplifications,
00:49:24.840 --> 00:49:30.390
but it's wonderful, because
we can use something we barely
00:49:30.390 --> 00:49:32.550
understand for the first time.
00:49:32.550 --> 00:49:34.590
And actually reach
that level of, yeah, I
00:49:34.590 --> 00:49:38.030
can understand
macroscopic behaviors too
00:49:38.030 --> 00:49:42.180
and how they relate to
quantum mechanical behavior
00:49:42.180 --> 00:49:43.760
of simple systems.
00:49:43.760 --> 00:49:45.115
OK, so that's where we're going.
00:49:45.115 --> 00:49:46.823
We're going to have
two or three lectures
00:49:46.823 --> 00:49:49.460
on harmonic oscillator.